Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $r = \dfrac{2p}{6p^2 - 10p} \div \dfrac{5}{10(3p - 5)} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{2p}{6p^2 - 10p} \times \dfrac{10(3p - 5)}{5} $ When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ 2p \times 10(3p - 5) } { (6p^2 - 10p) \times 5 } $ $ r = \dfrac {2p \times 10(3p - 5)} {5 \times 2p(3p - 5)} $ $ r = \dfrac{20p(3p - 5)}{10p(3p - 5)} $ We can cancel the $3p - 5$ so long as $3p - 5 \neq 0$ Therefore $p \neq \dfrac{5}{3}$ $r = \dfrac{20p \cancel{(3p - 5})}{10p \cancel{(3p - 5)}} = \dfrac{20p}{10p} = 2 $